2021, Vol. 2, Issue 1, Part A
Modular forms and their applications in modern number theory
Author(s): Kumar Aditay
Abstract: Modular forms constitute a cornerstone of modern number theory, bridging analytic, algebraic, and arithmetic domains through their symmetries and transformative properties. This paper provides an overview of modular forms, elucidating their foundational definitions, key constructions such as Eisenstein series and cusp forms, and their profound applications in resolving classical problems like the distribution of primes and the arithmetic of elliptic curves. Particular emphasis is placed on the Modularity Theorem, which asserts the modularity of elliptic curves over the rationals, and the role of modular forms in the Langlands programme. By synthesising historical developments with contemporary advancements, this analysis underscores modular forms' versatility in computational number theory and beyond, highlighting their enduring impact on Diophantine equations and automorphic representations.
DOI: https://doi.org/10.22271/math.2021.v2.i1a.260
Pages: 106-108 | Views: 203 | Downloads: 89
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How to cite this article:
Kumar Aditay. Modular forms and their applications in modern number theory. Journal of Mathematical Problems, Equations and Statistics. 2021; 2(1): 106-108. DOI: 10.22271/math.2021.v2.i1a.260
